$T\bar T$ and LST

0.24 MB·

by  Amit Giveon

¯
TT and LST
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Amit Giveon1, Nissan Itzhaki2 and David Kutasov3

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1Racah Institute of Physics, The Hebrew University
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t Jerusalem 91904, Israel
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e 2 Physics Department, Tel-Aviv University, Israel
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[ Ramat-Aviv, 69978, Israel

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3EFI and Department of Physics, University of Chicago
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5640 S. Ellis Av., Chicago, IL 60637, USA
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.
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0 Itwasrecentlyshownthatthetheoryobtainedbydeforming ageneraltwodimensional
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1 conformal theory by the irrelevant operator TT¯ is solvable. In the context of holography,
:
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a large class of such theories can be obtained by studying string theory on AdS . We show
i 3
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that a certain TT¯ deformation of the boundary CFT gives rise in the bulk to string theory
r 2
a
in a background that interpolates between AdS in the IR and a linear dilaton spacetime
3
in the UV, i.e. to a two dimensional vacuum of Little String Theory. This construction
provides holographic duals for a large class of vacua of string theory in asymptotically
linear dilaton spacetimes, and sheds light on the UV behavior of TT¯ deformed CFT . It
2
may provide a step towards holography in flat spacetime.
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1. Introduction
Vacua of quantum gravity in asymptotically anti de Sitter spacetimes are holograph-
ically dual to Quantum Field Theories (QFT’s) which approach a renormalization group
fixed point (a Conformal Field Theory) in the UV. This is reflected in the fact that the
entropy of the bulk theory, which is dominated at high energies by large AdS black holes,
grows as E like Eα, with α = d−2 < 1 (for AdS ), in agreement with the growth of
→ ∞ d−1 d
the entropy in a d 1 dimensional CFT.
−
One of the most important open problems in quantum gravity is to generalize the
highly successful AdS/CFT paradigm to other spacetimes, such as flat Minkowski space-
time IRd−1,1. There are many indications that holography plays a central role in such
spacetimes as well, but there is no useful description of the boundary theory. One way to
see the difficulty is to note that the entropy of very massive black holes in IRd−1,1 grows
like Eα with α = d−2 > 1, so the dual theory must exhibit this growth as well. This
d−3
implies that it cannot be a local QFT in the usual sense (an RG flow connecting two fixed
points).
It has been known for a long time that there is an interesting intermediate case,
Little String Theory (LST). From the bulk perspective, it corresponds to spacetimes of
the (asymptotic) form IR IRd−1,1 where IR is labeled by a radial coordinate φ, and has
φ φ
×
the property that the dilaton depends linearly on φ, at least near the boundary at φ ,
→ ∞
where the string coupling goes to zero. Such backgrounds arise naturally in string theory
near NS5-branes [1], and singularities of Calabi-Yau manifolds [2], and are believed to
exhibit holography as well [3].
The black holes that govern the high energy thermodynamics of LST have an entropy
that grows like the energy. In this sense, LST can be thought of as an intermediate case
between anti de Sitter spacetime, in which the entropy grows like Eα with α < 1, and flat
Minkowski spacetime, where α > 1. There are other ways in which this is the case. For
example, both linear dilatonand anti de Sitter spacetimes are solutions to (dilaton)gravity
with negative cosmological constant, while in flat spacetime the cosmological constant
vanishes. On the other hand, the time it takes signals to propagate from the bulk to the
boundary is finite in AdS spacetime and infinite in the other two.
All of the above suggests that understanding holography in asymptotically linear
dilatonspacetimes might provide a useful step towards holography in flat spacetime. What
makes this particularly interesting is that the Hagedorn behavior of the entropy at high
1
energies implies that the dual cannot be a standard local QFT, like in flat spacetime, and
it would be nice to understand it microscopically.
Most of the work on LST in the past treated it as a bulk theory in an asymptotically
linear dilaton spacetime, and there was no useful definition of the dual, or boundary,
theory. The aim of this note is to take advantage of recent progress in field theory [4,5]
to (partially) rectify this situation. We will argue that a large class of two dimensional
conformal field theories deformed by a certain TT¯ operator provide a boundary description
of two dimensional vacua of LST. In particular, they give a microscopic understanding of
the Hagedorn entropy of black holes in asymptotically linear dilaton backgrounds of the
form IR IR S1.
φ t
× ×
The plan of this note is the following. In section 2 we discuss some of the results of
[4,5]. In particular, we point out that the spectrum of a CFT on a spatial circle deformed
2
¯
by the irrelevant operator TT, interpolates between that of a CFT at low energies and
2
a spectrum with Hagedorn growth at high energies. In section 3, as preparation for a
discussion of the TT¯ deformation in holography, we review some elements of string theory
on AdS , and the LST that underlies it. In particular, we present a geometry that gives
3
rise to a bulk realization of an RG flow from a theory with Hagedorn density of states
in the UV to a CFT in the IR, which is qualitatively similar to that appearing in TT¯
2
deformed CFT .
2
In section 4 we discuss the TT¯ deformation of CFT in the context of holography. We
2
¯
show that a certain natural TT deformation corresponds in the bulk to a marginal current-
current deformation of the worldsheet theory on AdS . We investigate this deformation
3
in section 5, and find that it corresponds to string theory in the bulk geometry that
interpolates between AdS in the IR and a linear dilaton spacetime in the UV discussed in
3
section 3. Thus, it describes a particular (two dimensional) vacuum of LST. In section 6 we
compare the high energy thermodynamics of the boundary theory (a TT¯ deformed CFT )
2
to that of the bulk theory, obtained by studying black holes in the deformed background,
and find agreement between the two. In section 7 we comment on our results.
A note on terminology: when discussing the (deformed) CFT in the context of holog-
2
raphy, we will alternate between referring to it as a boundary (as opposed to bulk) and a
spacetime (as opposed to worldsheet) theory.
2
2. TT¯ deformation of CFT
2
In this section we discuss some aspects of the recent work [4,5] (see also [6,7]) on the
theory obtained by deforming the Lagrangian of a two dimensional conformal field theory
(CFT ) by
2
δ = tTT¯, (2.1)
L
where T and T¯ are the holomorphic and anti-holomorphic components of the stress tensor,
respectively, and the composite operator TT¯ is defined at finite coupling t as [8]
TT¯(y) = lim T(x)T¯(y) Θ(x)Θ(y) . (2.2)
x→y −
(cid:0) (cid:1)
Θ is the trace of the stress tensor, related to T and T¯ by the conservation equations
∂ T = ∂ Θ; ∂ T¯ = ∂ Θ. (2.3)
x¯ x x x¯
The operator TT¯ has dimension four. Hence the coupling t in (2.1) has units of length
squared; in particular, it is irrelevant (in the RG sense). The resulting theory approaches
the original CFT at long distances, whereas at short distances one in general expects the
theory to lose predictive power.
The authors of [4,5] used techniques from integrable field theory to study the defor-
mation (2.1). In particular, they calculated exactly the spectrum of the theory on a circle
of circumference R and found it to be
R R2 4π c
E(R,t) = + + (h ) , (2.4)
−2t r4t2 t − 24
where c is the central charge of the original CFT. The coupling t must be taken to be
positive for the theory to have a vacuum [4].
In the infrared limit t 0, (2.4) approaches the standard CFT result,
→
4π c
E(R,0) = (h ), (2.5)
R − 24
corresponding to a state1 with L = L¯ = h. Equation (2.4) describes the change of the
0 0
energy of such a state as we turn on the perturbation. A few comments about (2.4) are in
order at this point:
1 In (2.4) we exhibited the result for states with h = h¯. The generalization to the case h 6= h¯
is known.
3
(1) This equation relates the three dimensionless quantities that appear in the problem,
4πt ER c
b = , = , M = 2h . (2.6)
R2 E 2π − 12
In terms of these quantities, it takes the form
1 1 2M
(b,M) = + + . (2.7)
E −b rb2 b
(2) b in (2.6) can be thought of as the value of the coupling t (2.1) at the scale R. Small b
corresponds to weak coupling at distances R. In this weak coupling regime, there is
≥
a large number of states, those for which M 1/b, whose energies are little changed
| | ≪
by the perturbation and are given to a good approximation by (2.5), M. As Mb
E ≃ | |
grows, the deviations of the spectrum (2.7) from the CFT one increase. In particular,
for high energies (Mb 1) one finds
≫
2M 2πM
, or E . (2.8)
E ≃ r b ≃ r t
Interestingly, the spectrum becomes R independent in this limit; the energy scale is
set by the coupling t. Note that in the weak coupling regime b 1, the energy scale
≪
in (2.8), 1/√t, is much higher than the energy scale of the original CFT, which is of
order 1/R.
(3) The coupling plays an important role for low lying states above the SL(2,IR) invariant
ground stateas well. Consider, for example, the ground state itself, which corresponds
to h = 0, M = c/12, (2.6). As explained in the previous comment, the properties
−
of this state start to deviate significantly from the original CFT when bc becomes of
order one. In particular, for bc > 6 one finds that the energy of the ground state (2.7)
becomes complex. For c 1 this does not happen at weak coupling, but for large c
∼
it does. One can think of bc as a ‘t Hooft coupling for this theory. We will mostly
discuss the regime of weak coupling (b 1) and arbitrary ‘t Hooft coupling bc.
≪
(4) One can use the energy formula (2.7) to analyze the entropy of the deformed theory.
Since this formula relates each state in the theory (2.1) to a state in the original CFT,
the entropy S( ) is given by the usual Cardy form, S = S (M), with M expressed in
c
E
terms of via (2.7). The Cardy entropy of the CFT is
E
c
S (M) 2π M , (2.9)
c
≃ r3
4
valid for M 1. Using (2.7), this can be written as
≫
c
S( ) = 2π (2 +b 2) , (2.10)
E r6 E E
which is expected to be valid for 1. For 1 1/b, this reduces to the original
E ≫ ≪ E ≪
CFT entropy (2.9). In the opposite regime, 1/b one finds instead
E ≫
bc 2πct
S 2π = E . (2.11)
≃ r 6 E r 3
This is a Hagedorn entropy, S = β E, with inverse Hagedorn temperature
H
2πct
β = . (2.12)
H r 3
Note that β is equal to the circumference of the circle, R, at the point where the
H
ground state energy becomes complex (corresponding to bc = 6 in (2.6)). This is a
UV/IR relation reminiscent of the relation between the mass of the winding tachyon
and the Hagedorn temperature in (perturbative) critical string theory (see e.g. [9]).
Likethere, inmodelswithfermionsthestatewithM = c/12corresponds tofermions
−
with anti-periodic boundary conditions around the circle.
¯
(5) The fact that the TT deformed theory (2.1) has a Hagedorn density of states implies
thatitsUVbehaviorisnotgovernedbyanRGfixedpoint. Atheorywiththisbehavior
is not expected to have local operators, such as the stress tensor (2.3). Thus, some of
the assumptions that were used in [8] are not satisfied in this case. Nevertheless, the
results of [4,5] suggest that the analysis of [8] is still valid.
(6) There are RG flows in QFT that look in the vicinity of an IR fixed point as TT¯
deformations ofa CFT , but leadintheUV to a fixedpoint rather thanto a Hagedorn
2
spectrum (see e.g. [10,11]). The reason for the difference with the results of [4,5] is
¯
that in the latter case the theory looks like a TT deformation at all scales, and not
just in the IR.
We see that at high energies the theory (2.1) does not approach a UV fixed point, but
rather has a Hagedorn spectrum (2.11). There is a class of interacting non-gravitational
theories that is known to have a Hagedorn density of states at high energies – Little String
Theory. Thus, it is natural to ask whether the theories (2.1) give rise to two dimensional
vacua of LST. In the remainder of this note we will argue that in a large class of examples
this is indeed the case.
5
3. Aspects of the AdS /CFT correspondence
3 2
To study the results of [4,5] in the context of the AdS/CFT correspondence, we start
in this section with a brief review of some relevant aspects of string theory on AdS
3
(see e.g. [12,13] for more detailed discussions). We will take the AdS background to be
3
supported by a flux of the Neveu-Schwartz B-field. In this case the worldsheet theory on
AdS is solvable – it is described by the WZW model on the SL(2,IR) group manifold.
3
The worldsheet dynamics is governed to a large extent by a left-moving SL(2,IR) current
algebra generated by the currents Ja(z), a = 3,+, , and another, right-moving, current
−
algebra J¯a(z¯), both at level k. The level determines the size of the AdS space in string
units, R = √kl .
AdS s
A typical vacuum of string theory on AdS takes the form
3
AdS , (3.1)
3
×N
where is a compact CFT, whose central charge is determined by the condition that the
N
total worldsheet central charge of (3.1) is critical (twenty six for the bosonic string, and
fifteen forthesuperstring). Awellstudied exampleissuperstring theorywith = S3 T4,
N ×
which describes the near-horizon geometry of k NS5-branes wrapped on T4 S1 and p
×
fundamental strings wrapped on S1. The S3 corresponds to the angular directions in the
IR4 transverse to both the strings and the fivebranes, and is described by a WZW model
on the SU(2) group manifold. The total levels of SU(2) and SL(2,R) are both equal to k.
The worldsheet SL(2,IR) SL(2,IR) currents playan importantroleinthestudy of
L R
×
thetheory. Thezero modes ofthesecurrents giveconserved globalcharges inthespacetime
theory, which correspond to the global part of the conformal group of the spacetime (or
boundary)theory. Inparticular, J3 givesL0, whileJ± giveL±1, andsimilarlyfor theother
worldsheet and spacetime chirality. The full (anti) holomorphic stress tensor T(x) (T¯(x¯))
of the spacetime theory is constructed in terms of these currents and other observables on
AdS [13].
3
Anobservation[2]thatwillplayanimportantroleinourdiscussionbelowisthatvacua
of the form (3.1) are closely related to two dimensional vacua of LST. Indeed, consider
string theory in the background
IR IR S1 , (3.2)
t φ
× × ×N
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where the first factor is time, the second a spatial direction with linear dilaton along it,
and the rest as before. The slope of the linear dilaton, Q, is determined by the criticality
condition. As φ , the string coupling g exp( Qφ/2) goes to zero. φ is
s
→ ∞ ≃ − → ∞
the boundary of the spacetime (3.2), on which the dual theory lives. As φ the
→ −∞
coupling diverges. To study physics in this region requires some understanding of the
strong coupling behavior of the theory.
String theory in the background (3.2) is believed to be holographically dual to a non-
local, non-gravitational theory [3]. The high energy behavior of the entropy is dominated
by black holes, which are obtained by replacing IR IR in (3.2) by the two dimensional
t φ
×
black hole background. This black hole is described by an exactly solvable worldsheet
theory, which corresponds to the coset SL(2,IR)/U(1) [14]. The high energy behavior of
the entropy is Hagedorn, S(E) = β E, with
H
β = 2π√kl . (3.3)
H s
Some thermodynamic properties of this black hole were studied in [15,16].
The relation between (3.1) and (3.2) can be thought of as follows [2]. One starts
with the background (3.2) and adds to it p fundamental strings wrapping the S1. The
addition of the strings modifies the background in the IR (i.e. in the region φ ). In
→ −∞
particular, the string coupling no longer grows without bound there; rather it saturates at
a value g2 1/p. Thus, for large p the coupling is small everywhere, and one can study
s ∼
bulk string theory in this background using perturbative techniques.
The full background takes in this case the form [2] (see also [17])
, (3.4)
3
M ×N
where is as in (3.1), (3.2), and the three dimensional background is
3
N M
ds2 =f−1l2dγdγ¯+kl2dφ2,
1 s s
v
e2Φ = e−2φf−1, (3.5)
p 1
dB =2ie−2φf−1ǫ ,
1 3
where l γ = x1 +x0, l γ¯ = x1 x0,
s s
−
1
f = 1+ e−2φ, (3.6)
1
k
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and v is a constant associated with the compact CFT . Recall that x1 is periodically
N
identified, x1 x1 +R.
∼
The background (3.4), (3.5) interpolates between the LST background (3.2) near the
boundary at φ and the AdS background (3.1) in the infrared φ . It describes
3
→ ∞ → −∞
an RG flow from a non-local theory with a Hagedorn spectrum in the UV, to a standard
CFT in the IR. The transition between the two occurs at a value of φ determined by the
2
requirement that the two terms in f (3.6) are comparable. We can choose this scale to
1
take any convenient value, and this has effectively been done in (3.6). The RG flow (3.5)
will be the focus of our discussion below.
In the example mentioned above, where = S3 T4, the UV geometry (3.2) is
N ×
the CHS geometry [1] of k NS5-branes wrapped on T4 S1, and the non-local theory
×
in question is the six dimensional LST of k fivebranes compactified on a torus. The RG
flow (3.5) interpolates in this case between this non-local theory in the UV and the CFT
2
obtainedby studying thesystemofk NS5-branes andpfundamental stringsintheinfrared
limit, which is dual to string theory on AdS S3 T4.
3
× ×
Another observation which will play a role below concerns the structure of the bound-
ary CFT corresponding to the background (3.1). It has been proposed [18,19] that at large
p this theory has the form of a symmetric product
p/S , (3.7)
p
M
where is a CFT with central charge cM = 6k. Roughly speaking, can be thought
M M
of as the CFT associated with a single string added to the background (3.2), and the
structure (3.7) relies on the fact that at large p the interaction between the p strings in
the background (3.2) goes to zero. The status of (3.7) is not completely clear, but we will
assume it below. Our results provide some further support for this assumption.
Since for large p string theory in the background (3.1) is weakly coupled, one can use
perturbative (worldsheet) techniques to study it. In particular, one can construct vertex
operators that describe low lying local operators in the boundary theory [12,13]. In terms
of(3.7)one can thinkof these vertex operatorsas“singletrace” operatorsin thesymmetric
orbifold CFT. More precisely, they correspond to operators of the form
p
(x), (3.8)
i
O
Xi=1
where correspond to a particular operator (x) living in the i’th factor in (3.7),
i
O O ∈ M
with the sum over i imposing S invariance.
p
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4. The TT¯ deformation in AdS
3
¯
In order to make contact with the discussion of [4,5], we need to construct the TT
deformation in string theory on AdS . One possibility is to use the vertex operator of the
3
stress tensor in [13], whose construction we will review shortly, and consider the double
trace deformation by the product of the vertex operators for T and T¯. We will next argue
that there is also a single trace TT¯ deformation, which is easier to study in the bulk.
To understand it, it is useful to start with a brief review of the construction of the
stress tensor in [13].2 There are two observables that play a key role in this construction.
One is the current
J(x;z) = 2xJ (z) J+(z) x2J−(z), (4.1)
3
− −
whichcombinestheSL(2,IR) currentsintoasingleobjectlabeledbytheauxiliaryvariable
L
x. This variable can be thought of as position on the boundary of AdS . There is also a
3
right-moving analog of (4.1) with the opposite worldsheet and spacetime chirality J¯(x¯;z¯).
The second observable is
2h
1 1
Φ (x;z) = , (4.2)
h π (cid:18) γ x 2eφ +e−φ(cid:19)
| − |
which is an eigenfunction of the Laplacian on AdS , and gives rise in the quantum theory
3
to a primary of the worldsheet and spacetime Virasoro, with worldsheet dimension ∆ =
h
∆¯ = h(h 1)/(k 2) and spacetime dimension (h,h).
h
− − −
Intermsoftheoperators(4.1),(4.2), thevertexoperatorofthespacetimestress-tensor
is given by
1
T(x) = d2z(∂ J∂ Φ +2∂2JΦ )J¯(x¯;z¯). (4.3)
2k Z x x 1 x 1
As explained in [13], this operator is physical (i.e. BRST invariant), its spacetime scaling
dimension is (2,0), it is holomorphic, ∂ T = 0 (as expected from the unitarity of the
x¯
spacetime CFT), and it satisfies the standard OPE algebra of the holomorphic stress-
tensor in the spacetime CFT . The anti-holomorphic stress tensor T¯(x¯) is constructed
2
similarly by flipping all the chiralities in (4.3), (x,z,J) (x¯,z¯,J¯).
↔
The TT¯ deformation of [4,5] corresponds in terms of the above discussion to adding
to the worldsheet action the product of the vertex operators for T and T¯. Since each of
2 For simplicity, we will discuss the construction in the bosonic string. The generalization to
the superstring is explained in [13].
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